The usual second isomorphism theorem for groups is: let $G$ be a group, $S$ and $N$ subgroups with $N$ normal, then $SN$ is a subgroup of $G$, $S\cap N$ is a normal subgroup of $S$ and $SN/N \simeq S /S \cap N$.

($N$ need not to be a normal subgroup as long as $S$ is a subgroup of the normalizer of $N$).

Now if $S$ **and** $N$ are normal subgroups of $G$ then $SN/N \simeq S /S \cap N$ **and** $SN/S \simeq N /S \cap N$, so that the normal chains $(S \cap N \subset S \subset SN)$ and $(S \cap N \subset N \subset SN)$ are equivalent.

It's this last property that we would like to generalize to the inclusions of groups $(H \subset G)$.

The notion of *normal* subgroup $K \triangleleft G$ (i.e. $ \forall g \in G \text{ , } Kg=gK $) can be generalized by the notion of *normal intermediate* subgroup (motivated by the prop.3.3 p476 of this paper).

**Definition**: $K$ is a **normal intermediate** subgroup of the inclusion $(H \subset G)$ if $H \subset K \subset G$, and $$\forall g \in G \text{ , } KgH=HgK $$
**Remark**: If $K_i$ is a normal intermediate subgroup of $(H \subset G)$, then so are $K_1 \cap K_2$ and $K_1K_2$,

and if $H \subset H_0 \subset K_i \subset G_0 \subset G$ then $K_i$ is also a normal intermediate subgroup of $(H_0 \subset G_0)$.

**Definition**: A chain of subgroups $(K_1 \subset K_2 \subset \dots \subset K_n)$ is a *normal intermediate* chain if $K_i$ is a normal intermediate subgroup of the inclusion $(K_{i-1} \subset K_{i+1})$.

**Definition**: Two chains of subgroups $(K_1 \subset K_2 \subset \dots \subset K_n)$ and $(L_1 \subset L_2 \subset \dots \subset L_m)$ are equivalent if $m=n$ and $\exists \sigma \in S_{n-1}$ such that $(K_i \subset K_{i+1}) \sim (L_{\sigma(i)} \subset L_{\sigma(i)+1})$.

**Definition**: Two inclusions of groups are equivalent, $(A \subset B) \sim (C \subset D)$, if: $(A/A_B \subset B/A_B) \simeq (C/C_D \subset D/C_D)$ with $A_B$ the normal core of $A$ in $B$.

Question: Let $K$ and $L$ be normal intermediate subgroups of an inclusion $(H \subset G)$ then, are thenormal intermediatechains $(K \cap L \subset K \subset KL)$ and $(K \cap L \subset L \subset KL)$ equivalent ?

**Examples**: See this post and note that this answer gives here an example.

**Motivation**: A Jordan-Hölder theorem generalized to the inclusions of groups.